1

Systems of Linear Equations

• 4-1 Systems of Linear Equations in Two Variables

2

4-1 Systems of Linear Equations in Two Variables

Deciding whether an ordered pair is a solution of a linear system. The solution set of a linear system of equations contains all ordered pairs that satisfy all the equations at the same time.

• Example 1: Is the ordered pair a solution of the given system? 2x + y = -6 Substitute the ordered pair into each equation.

x + 3y = 2 Both equations must be satisfied.

A) (-4, 2) B) (3, -12)2(-4) + 2 = -6 2(3) + (-12) = -6(-4) + 3(2) = 2 (3) + 3(-12) = 2

-6 = -6 -6 = -6 2 = 2 -33 ≠ -6

∴ Yes ∴ No

3

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems by Graphing. One way to find the solution set of a linear system of equations is to graph each equation and find the point where the graphs intersect.

• Example 1: Solve the system of equations by graphing. A) x + y = 5 B) 2x + y = -5

2x - y = 4 -x + 3y = 6

Solution: {(3,2)} Solution: {(-3,1)}

4

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems by Graphing. There are three possible solutions to a system of linear equations in two variables that have been graphed:

• 1) The two graphs intersect at a single point. The coordinates give the solution of the system. In this case, the solution is “consistent” and the equations are “independent”.

• 2) The graphs are parallel lines. (Slopes are equal) In this case the system is “inconsistent” and the solution set is 0 or null.

• 3) The graphs are the same line. (Slopes and y-intercepts are the same) In this case, the equations are “dependent” and the solution set is an infinite set of ordered pairs.

5

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Elimination. Remember: If a=b and c=d, then a + c = b + d.

Step 1: Write both equations in standard form

Step 2: Make the coefficients of one pair of variable terms opposite

(Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either x or y will be zero.)

Step 3: Add the new equations to eliminate a variable

Step 4: Solve the equation formed in step 3

Step 5: Substitute the result of Step 4 into either of the original equations and solve for the other value.

Step 6: Check the solution and write the solution set.

6

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Elimination.

• Example 2: Solve the system : 2x + 3y = 19Step 1: Both equations are in standard form 3x - 7y = -6

Step 2: Choose the variable x to eliminate: Multiply the top equation by 3, the bottom equation by -2

3[2x + 3y = 19] 6x + 9y = 57 -2[3x - 7y = -6] -6x +14y = 12

Step 3: Add the new equations to eliminate a variable

0x + 23y = 69Step 4: Solve the equation formed in step y = 3

Step 5: Substitute the result of Step 4 into either of the original equations and solve for the other value. 2x + 3(3) = 19

2x = 10 x = 5 Solution Set: {(5,3)}

Step 6: Check the solution and write the solution set.

7

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Elimination.

• Example 3:

Solve the system :

2[2x - 3y = 1] 4x - 6y = 2

-3[3x - 2y = 9] -9x + 6y = -27

-5x + 0y = -25 x = 5 3(5) - 2y = 9

-2y = -6Solution Set: {(5,3)} y = 3

1 1 1 rewrite as 6[ ] 2 3 1

3 2 6

: 2 3 1

1 1 1

3 2 63 2 9

3 2 9

x y x y

Solv

x y

x y

e x y

x y

⇒ − = ⇒ − =

− =−

− =

=

=

−

8

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Elimination.

• Example 4: Solve the system : 2x + y = 6

-8x - 4y = -24

4[2x + y = 6] 8x + 4y = 24

-8x -4y = -24 -8x - 4y = -24

0 = 0 True

Solution Set: {(x,y)| 2x + y = 6}

Note: When a system has dependent equations and an infinite number of solutions, either equation can be used to produce the solution set. Answer is given in set-builder notation.

9

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Elimination.

• Example 5: Solve the system : 4x - 3y = 8

8x - 6y = 14

-2[4x - 3y = 8] -8x + 6y = -16

8x - 6y = 14 8x - 6y = 24

0 = 8 False

Solution Set: 0 or null

Note: There are no ordered pairs that satisfy both equations. The lines are parallel. There is no solution.

10

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Substitution.

Step 1: Solve one of the equations for either variable

Step 2: Substitute for that variable in the other equation

(The result should be an equation with just one variable)

Step 3: Solve the equation from step 2

Step 4: Substitute the result of Step 3 into either of the original equations and solve for the other value.

Step 6: Check the solution and write the solution set.

11

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Substitution.

• Example 6: Solve the system : 4x + y = 52x - 3y =13

Step 1: Choose the variable y to solve for in the top equation:

y = -4x + 5Step 2: Substitute this variable into the bottom equation

2x - 3(-4x + 5) = 13 2x + 12x - 15 = 13Step 3: Solve the equation formed in step 2

14x = 28 x = 2Step 4: Substitute the result of Step 3 into either of the original equations and solve for the other value. 4(2) + y = 5

y = -3Solution Set: {(2,-3)}

Step 5: Check the solution and write the solution set.

12

4-1 Systems of Linear Equations in Two Variables

Solving Linear Systems of two variables by Method of Substitution.

• Example 7:

Solve the system :

y = -2x + 2

-2x + 5(-2x + 2) = 22 -2x - 10x + 10 = 22

-12x = 12 x = -1 2(-1) + y = 2

y = 4Solution Set: {(-1,4)}

1 1 1

2 4 22

1 1 1 rewrite as 4[ ] 2 2

2 4 2

: 2 2

-2 5 2

5 2

2

2

x y

x y

x y x y

Solve x y

x y

⇒ + = ⇒ + =

++

=+

=

=

=

+

−